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Let $D^n$ be the closed unit disk in $\mathbb{R}^n$. I'm looking for a homeomorphism of $D^n$ to itself which is the identity on the boundary, and takes the origin to a given point $z$ in its interior. I can visualize such a function as "shifting around the mass" within $D^n$, but I can't seem to write one down.

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up vote 4 down vote accepted

Map the rays starting at $z$ to the rays starting at the origin. Each ray is identified by its point on the boundary.

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for a unit vector $x$ and $t\in[0,1]$ you can use $tx\mapsto z+t(x-z$) – yoyo Mar 1 '11 at 15:45

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